Physical Chemistry Third Edition

690 16 The Principles of Quantum Mechanics. II. The Postulates of Quantum Mechanics

The operator for|L|, the magnitude of the angular momentum, is not usable

since

|L|(L^2 )^1 /^2 (L^2 x+L^2 y+L^2 z)^1 /^2 (16.3-15)

and we have no procedure for unscrambling the square root when the operator is

formed. To obtain information about the magnitude of the angular momentum we will

use theL^2 operator.

Exercise 16.3

a.Construct the operator forL̂x.

b.Construct the operator forL̂y.

c.Construct the operator forL^2 z(the square of the operator forLz).

It is sometimes useful to have operator expressions for commutators.

EXAMPLE16.4

Find the operator equal to the commutator

[

̂x,p̂x

]

.

Solution

Operate on an arbitrary differentiable functionf(x):

[

̂x,̂px

]

f ̄

h

i

[

x

∂f

∂x

−

∂(xf)

∂x

]

− ̄

h

i

f

The operator equation is

[

̂x,p̂x

]

− ̄

h

i

ih ̄ (16.3-16)

Equation (16.3-16) is an important result. Some authors find the form of the operator

̂pxby postulating that this commutation relation must hold.

Exercise 16.4

a.Find the commutator [̂px,̂py].

b.Show that [

̂Lx,̂py

]

ih ̄̂pz (16.3-17)

If an operator is needed in another coordinate system, the operator is first constructed

in Cartesian coordinates and then transformed to the other coordinate system. For

example, if a particle moves in thexyplane, its position can be represented using the

polar coordinatesρandφ:

ρ

√

x^2 +y^2 (16.3-18)

φarctan(y/x) (16.3-19)